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Orbital Eccentricity Distribution of Solar-Neighbour Halo Stars

We present theoretical calculations for the differential distribution of stellar orbital eccentricity for a sample of solar-neighbour halo stars. Two types of static, spherical gravitational potentials are adopted to define the eccentricity e for given energy E and angular momentum L, such as an isochrone potential and a Navarro-Frenk-White potential that can serve as two extreme ends covering in-between any realistic potential of the Milky Way halo. The solar-neighbour eccentricity distribution ΔN(e) is then formulated, based on a static distribution function of the form f(E,L) in which the velocity anisotropy parameter βmonotonically increases in the radial direction away from the galaxy center, such that beta is below unity (near isotropic velocity dispersion) in the central region and asymptotically approaches \sim 1 (radially anisotropic velocity dispersion) in the far distant region of the halo. We find that ΔN(e) sensitively depends upon the radial profile of β, and this sensitivity is used to constrain such profile in comparison with some observational properties of ΔN_{obs}(e) recently reported by Carollo et al. (2010). Especially, the linear e-distribution and the fraction of higher-e stars for their sample of solar-neighbour inner-halo stars rule out a constant profile of β, contrary to the opposite claim by Bond et al. (2010). Our constraint of β\lesssim 0.5 at the galaxy center indicates that the violent relaxation that has acted on the inner halo is effective within a scale radius of \sim 10 kpc from the galaxy center. We discuss that our result would help understand the formation and evolution of the Milky Way halo.